Problem 13 - Basic Engineering Mathematics

Question

An automatic garden spray produces a spray to a distance of \(1.8 \mathrm{~m}\) and revolves through an angle \(\alpha\) which may be varied. If the desired spray catchment area is to be \(2.5 \mathrm{~m}^{2}\), to what should angle \(\alpha\) be set, correct to the nearest degree.

Step-by-Step Solution

Answer

The angle \( \alpha \) should be set to 88 degrees.

Step 1: Understanding the Problem

We need to find the angle \( \alpha \) that makes the sector area of a circle with radius \( 1.8 \) meters equal to \( 2.5 \) square meters. The spray creates a sector of a circle, and the area of a sector is given by \( A = \frac{1}{2} r^{2} \alpha \), where \( \alpha \) is in radians.

Step 2: Setting Up the Equation

The radius \( r \) of the spray is \( 1.8 \) meters, and the area \( A \) is \( 2.5 \) square meters. Plug these values into the formula for the area of a sector: \[ 2.5 = \frac{1}{2} \times (1.8)^2 \times \alpha \].

Step 3: Simplifying the Equation

Calculate \( (1.8)^2 = 3.24 \). Substitute this value into the equation: \[ 2.5 = \frac{1}{2} \times 3.24 \times \alpha \]. This simplifies to \[ 2.5 = 1.62 \times \alpha \].

Step 4: Solving for \(\alpha\) in Radians

To find \( \alpha \), divide both sides of the equation by \( 1.62 \): \[ \alpha = \frac{2.5}{1.62} \approx 1.5432 \text{ radians} \].

Step 5: Converting Radians to Degrees

Since the angle \( \alpha \) needs to be in degrees, convert from radians by using the conversion factor \( 180/\pi \). Therefore, \[ \alpha \approx 1.5432 \times \frac{180}{\pi} \approx 88.4^\circ \].

Step 6: Rounding to the Nearest Degree

Since \( \alpha \approx 88.4^\circ \), we round to the nearest degree to get \( \alpha = 88^\circ \).

Key Concepts

Circle Sector Area

In geometry, the concept of a circle sector is like slicing a piece of pie from a whole circle. The area of a circle sector is the space enclosed by the two radii and the arc of a circle. To calculate it, you use the formula:

• \( A = \frac{1}{2} r^{2} \alpha \)

where:

• \( A \) is the area of the sector
• \( r \) is the radius of the circle
• \( \alpha \) is the central angle of the sector in radians

So if you know the radius of the circle and the angle, you can compute how much of the circle's area is taken up by the sector.
Using radians simplifies calculations because they directly relate an angle to the radius and the arc length. It's essential for students to understand this relationship so they can apply it to various geometry problems effectively.

Radian to Degree Conversion

Sometimes we need to convert angles from radians to degrees, especially if degrees are required by the problem or more intuitive to understand. The conversion is straightforward:

• The formula is \( \,\text{Degrees} = \,\text{Radians} \times \frac{180}{\pi} \)

This conversion works because \( \pi \) radians is the angle for a half-circle, which is equivalent to 180 degrees. When you multiply radians by this fraction, you're essentially scaling the angle to fit the full rotation measure most used in everyday contexts.
Angling measurements in degrees is common for applications like navigation, while radians might be more practical in higher mathematics and physics, where calculations often involve calculus and trigonometric identities.

Problem-Solving Steps

Problem-solving in geometry often involves systematic steps. To solve a sector area problem effectively, follow these simple steps:

• Understand what the problem is asking. Read it thoroughly.
• Identify the known quantities. In our example, this includes the radius and the area of the sector.
• Set up your equation based on the formula for the area of a sector.
• Simplify the equation by substituting known values and performing any necessary algebraic operations.
• Solve for the unknown variable, such as the angle in radians.
• If asked, convert your result into a more familiar unit or form, like degrees.
• Lastly, double-check your calculations for any errors or missteps.

Following these steps with precision helps ensure that nothing is overlooked, leading to a correct solution. Approach each geometry problem with patience, and don't rush through the process.